1. Field of the Invention
The present invention relates to the coherent sampling method and apparatus.
The invention is particularly concerned with a digital sampling oscilloscope (DSO) employing the coherent sampling to measure input repetitive signals with fine time resolution.
The novel digital oscilloscope can acquire waveforms without waveform missing phenomena for a short time period by means of a equivalent time sampling.
2. Description of the Prior Art
The equivalent time sampling is widely employed in the digital oscilloscopes. The sampling is well known as the measures to obtain waveforms sampled with the finer time resolution than that of the time period of the sampling clock.
The equivalent time sampling includes three systems. The first is the sequential sampling. The second is the random sampling. The third is the coherent sampling.
The prior art of digital oscilloscopes and their related technics are disclosed as follows.
Prior Art 1; Picosecond Domain Waveform Measurements, N. S. Nahman, Time-Domain Measurements in Electromagnetics, Van Nostrand Reinhold
Prior Art 2; IEEE Standard for Digitizing Waveform Recorders, IEEE Std 1057-1994 pp. pp. 5 & 28
Prior Art 3; U.S. Pat. No. 5,708,432, Jan. 13, 1998, Coherent Sampling Digitizer System, Reynolds et al.
Prior Art 4; Electrical Test Instruments, --Theory and Applications--R. A. Witte RTR Prentice Hall, pp. 120-121
Prior Art 5; The Microwave Transition Analyzer; A new Instrument Architecture for Component and Signal Analysis, D. J. Ballo and J. A. Wendler, October 1992, Hewlett-Packard Journal
Prior Art 6; Japanese Provisional Publication No. 10-293140 Nov. 4, 1998, Random Sampling Holdoff Method and Circuit, Uchida et al.
Prior Art 7; Waveform Missing Mechanisms and a Countermeasure in a Random Sampling System, IEEE Instrumentation and Measurement Technology conference, St. Paul Minnesota, U.S.A. May 18-21, 1998. K. Uchida et al.
Prior Art 8; Acquisition Clock Dithering in a Digital Oscilloscope, D. E. Tpeppen, April 1997, Hewlett-Packard Journal
In the prior art 1, the sequential sampling and the random sampling are disclosed.
In FIG. 1, there is shown a circuit of the sequential sampling disclosed in the prior art 1. In the sequential method, the signal f(t) of the recurrent pulse generator 81 is passed through a delay line DL to allow time for the sampling pulse to be generated. On successive signal occurrences, the delay generator 82 shifts the sampling time in a known a priori way, usually uniformly. The successive samples are stored in the signal (vertical) channel f(t) memory 84. The memory output 85 from the memory 84 is displayed in a order as indicated by the numbers at the points. No signal channel delay line may be required for triggerable signal sources if satisfactory electronic delay is available.
In FIG. 2, there is shown a circuit of the random sampling disclosed by the prior art 1. In the random method, the time value of each sample is not known a priori but is determined by measuring the relative time position (time point) between the start of the signal f(t) and the sampling pulse. The signal f(t) is generated by a recurrent pulse generator 81. The value so determined is stored in the time-base memory 87. Note that (1) the sampling pulse is not synchronized to the signal, (2) no signal delay line is thus required, and (3) non zero samples are obtained whenever the signal and sampling pulse occur simultaneously. The time ramp 86 could just as well be started by the signal f(t) and stopped by the sampling pulse 90.
The sequential sampling shown in FIG. 1 requires the delay line DL to obtain samples at trigger points or theretofore. The wider the bandwidth of the delay line DL is, the thinner it's diameter is. The thin delay line has the high cutoff frequency, however, the line is accompanied with much dissipation in a high frequency range. The bandwidth is, therefore, compromisingly limited by employing the delay line.
In order to obtain the fine time resolution of time in the operation of the sampling data acquisition, it is required to employ the time base with the fine time resolution and the wide bandwidth feature together. If no pretrigger signal is obtainable in the sequential sampling, the bandwidth and the fine time resolution are compromised because of the delay line.
In FIG. 2, the time ramp 86 is started by the start of the signal f(t) and stopped by the sampling pulse a(t). Thereby the outputs 85 of the memory 84 are displayed at the points indicated by the outputs n of the time base memory 87 in a random order as shown by the numbers from 1 to 13. In this way, the signal f(t) is reproduced as f(n).
The random sampling shown in FIG. 2 has the pretrigger ability to be able to sampling at points previous to the start of the signal f)t) to be measured. Accordingly, the random sampling requires no delay line which limits the bandwidth.
In the paragraph 4.1.5 of the prior art 2, the coherent sampling is disclosed as one of the equivalent time sampling. The coherent sampling is realized by setting the input signal's repetition frequency to that of sampling clock appropriately.
In the prior art 3, another coherent sampling is disclosed. Therein, the repetition frequency Fs of the sampling clock is set appropriately to the input signal's repetition frequency Ft in contrast with the prior art 2. The prior art 3 shows the condition to realized the coherent sampling. In the conditions, Ft/Fs=M/N, in which M and N are relatively prime integers. The integer N is the number of samples during a repetition cycle of the input waveform to be measured. The integer M is the number of cycles of the input waveform to produce data of N different time points representing one cycle of the input waveform.
In the prior arts 2 and 3, the coherent sampling is realized by means of appropriately setting one repetition frequency to another between the input signal and the sampling clock.
In the prior art 4, the random sampling is described as follows.
Since random repetitive sampling provides pretrigger information, it has largely displaced sequential sampling, except at microwave frequencies. At microwave frequencies, the time/division setting on the scope can be very small, causing the window of time that is viewed on the display to also be very small (perhaps 100 ps). The probability of a randomly acquired sample falling into the desired time window is so small that random repetitive sampling would take a long time to acquire the entire waveform. Alternatively, sequential sampling forces the sample points to occur within the desired time window so the entire waveform can be acquired quickly.
In the sequential sampling, the sampling frequency depends on the input signal's repetition frequency. When the repetition frequency becomes over several hundred kHz, the sampling frequency Fs of 100 kHz or so is generally employed. The sequential sampling's data acquisition time Tseq is given by EQU Tseq=(Tw/Tres)(1/Fs) (1)
in which Fs is the sampling frequency, Tw is the time window and Tres is the time resolution.
The equation (1) means, for example, that the acquisition time EQU Tseq=(100 ps/1 ps)(10 .mu.s)=1 ms
in which the time window Tw is 100 ps, the time resolution Tres is 1 ps, and the sampling frequency Fs is 100 kHz or 1/10 .mu.s.
In the random sampling, the data acquisition time Tran is given by EQU Tran={(Tmh)/(FsTw)}(Tw/Tres)k
or EQU Tran={(Tmh)/(FsTres)}k (2)
The Tmh is the hold time duration, (1/(FsTw)) is the reciprocal of the probability to sample the signal within the time window Tw by the sampling clock. The (Tw/Tres) is the number of acquisition data.
The reading, writing and other processes are executed during the hold time duration Tmh. The constant k is approximately given by 21 og(Tw/Tres)+1, which depends on the sampling uniformity. Because, the sampling are not executed uniformly for the limited time period. At arbitrary two points on signal, for example, one point can be sample many times, however, another can not be sampled. In case of Tw/Tres=100, k takes a value between 4 and 6, typically 5.
A conventional wideband random sampling oscilloscope operates with a clock rate Fs of 40 MHz. Let it be supposed in FIG. 2, that the clock cycle time Tc is (1/Fs) or 25 ns, the hold time duration Tmh is 100 clock's period of 2.5 .mu.s, the time resolution Tres is 1 PS, the time window Tw is 100 ps and the constant k is 4. Then, the random sampling data acquisition time period Tran (=2.5 .mu.s(25 ns/1 ps)k) is approximately 250 ms.
The sequential sampling frequency Fs is 100 kHz and the random sampling rate Fs is 40 MHz.
Accordingly the random sampling rate is 400 times (40 MHz/100 kHz) higher than that of the sequential sampling. The data acquisition time periods Tran and Tseq in the random sampling and the sequential sampling are 250 ms and 1 ms respectively. The time period Tran is 200 times or more longer than Tseq.
A conventional digital storage oscilloscope (DSO) employs a time interpolator of the dual slope type. The time interpolator requires, for example, 25000 (=25 ns/1 ps) clocks (Tmh=(25000)(25 ns)=625 .mu.s) in order to obtain a time resolution Tres of 1 ps by using a sampling rate Fs of 40 MHz (1Fs=25 ns). Then, the random sampling requires a very long data acquisition time period Tran of about 60 s. The time period Tran of 60 s is too long to use it practically.
In the coherent sampling shown in the prior art 3, there is the relationship given by EQU Ft/Fs=M/N
in which M and N are relatively prime integers, Ft is the input signal's repetition frequency, Fs is the sampling clock rate, N is the number of samples during a cycle of the input waveform to be measured, and M is the number of cycles of the input waveform to produce data of N different time points representing one cycle of the input waveform. The N is shown by EQU N=1/(FtTres)
in which the Tres is the time resolution. Therefore, the coherent sampling's data acquisition time period Tcoh is obtained by EQU Tcoh=M/Ft=N/Fs=1/(FsFtTres) (3)
in which Tcho does not include a measuring time period of the input signal's repetition frequency Ft and a setting time period of the clock rate.
Let it be supposed that the coherent sampling clock rate Fs is 40 HMz (1/Fs=25 ns), the input signal's repetition frequency Ft is 1 GHz (1/Ft=1 ns), and the time resolution Tres is 1 ps. Then the coherent sampling data acquisition time Tcoh is given by EQU Tcoh=25 ns(1 ns/1 ps)=0.025 ms
In spite of the coherent sampling clock rate Fs is the same as that of the random, Tcoh (=0.025 ms) is 1/10000 times shorter than the random sampling's data acquisition time Tran (=250 ms). When the input signal's repetition frequency Ft is 10 GHz, the time Tcoh becomes 2.5 .mu.s. If the sequential sampling frequency Fs of 40 MHz is available, the sequential sampling data acquisition time Tseq of 2.5 .mu.s being equal to Tcoh can be obtained.
In the coherent sampling, the sampling clock rate is sought, and then different time positions on the input waveform are successively sampled at the sampling clock rate. In the sequential sampling, the required portion of the input waveform after the trigger are sampled. In the coherent sampling, however, after sampling one cycle of the input waveform, the required portion of the waveform is displayed.
The prior art 5 describes, in its page 5, the case to sample coherently a microwave signal as follows.
The signal at the IF is a replicas of the input signal, but at a much lower fundamental frequency. When this signal is digitized and displayed, the wave shape matches that of the input.
To keep the display triggered, low-frequency trigger circuitry is connected to the IF signal and used to initiate the storage of a data record relative to a rising or falling edge. Data samples in the buffer before the trigger occurrence are displayed as negative time (pretrigger view). Through the combination of periodic sampling and a low-frequency trigger circuit, the microwave transition analyzer is able to trigger internally on periodic signals across the full 40-GHz input bandwidth and offer negative-time capability without delay lines.
As above-mentioned, the coherent sampling is an excellent high frequency waveform data acquisition system with the pre-trigger ability.
To sample coherently, it is the premise that fluctuations of the signal to be measured and the sampling clock rate are negligible small. The premise does not exist in the sequential or random sampling. It is, therefore, the new restriction.
FIGS. 3, 4, 5 and 6 show waveforms reproduced by simulation of the coherent sampling and the fixed clock random sampling, of which X and Y coordinates are respectively the time t and the signal amplitude A.
In those FIGS. 3 to 6, it is assumed that a mean time period of fluctuated signal Tbase=10 and the standard deviation of the fluctuation FL=0.01 and 0.003. FIGS. 3 and 4 show the cases of the coherent sampling. FIGS. 5 and 6 show the cases of the random sampling.
The simulation is executed under the following conditions. The time resolution Tres is Tc/64. Data acquisitions are executed 640 times, i.e., 10 times per a time resolution Tres. The sampling clock cycle time Tc is 10(1+1/64) with no fluctuation of Tc.
Here assumed that the accumulated fluctuation of the signal increases with a square root of the lapse of time (.delta. t=(rndm)t.sup.1/2) and is given by EQU signal (t)=Sin[2.pi.{t+(rndm)t.sup.1/2 }/Tbase] (4)
Therein, rndm=(FL)Random []Tbase.sup.1/2 in which Random [] means to generate the Gaussian random numbers with its variance of 1 and its mean value of zero.
In the random sampling of FIGS. 5 and 6, there are quantizing errors depending on the time resolution of Tc/64. However, the jitters at the rising edge of the triggered time (equivalent time t=0) are decreased to zero. On the other hand, in the coherent sampling of FIGS. 3 and 4, cyclic fluctuation appear depending no the starndard deviation FLs and the jitters are observable over the whole waveforms. It is, therefore, difficult to measure a rise time or a settling time of the acquired waveform data. If data processing like averaging is applied to acquired data, there occur problems of degradation of a bandwidth, a time resolution and so on. The problems are to be solved.
The outlines of the three systems of the equivalent-time sequential, random and coherent sampling, and their advantages and disadvantages have been described. They can reach a compromise. The sequential sampling includes a barrier against the wide bandwidth. The coherent sampling involves a demerit of the time jitter occurrence. And the random sampling has a subject of the long data acquisition time.
In accordance with computers or digital circuits operating with high clock rate, a convenient realtime DSO operates with the sampling clock rate of 5 GHz or so. In the random sampling, let assume that a sampling clock rate Fs is 1 GHz, a hold time duration Tmh is 10 times period (10 ns) of the clock cycle time (1 Fs=1 ns), a time window Tw is 100 ps and a time resolution Tres is 1 ps. A random sampling data acquisition time Tram will be given by EQU Tran=10 ns(1 ns/100 ps)(100 ps/1 ps)k=0.04 ms
in which k=4. The above-assumption is easily realizable by the current technology. However, to obtain a shorter time period Tran, the random sampling must be confronted by the peculiar phenomena of waveform missing.
The prior arts 6 and 7 describe about the waveform missing phenomena as follows.
When the sampling clock rate is constant, one of the waveform missing phenomena is caused by the holdoff.
The prior arts 6 and 7 detailedly describe the mechanism causing the phenomena and the relationship between the random holdoff and the constant holdoff as dealing method with the phenomena. Therein, some problems have been solved.
There is, however, an unsolved problem. The problem is the waveform missing phenomena caused by a relation of a signal's repetition frequency Ft and a sampling clock rate Fs. When an input signal's repetition frequency Ft equals harmonics or subharmonics of a sampling clock rate Fs, namely, on one of the coherent states, specific time points on the waveform are repeatedly sampled. The operation of data acquisition over the whole waveform cannot be, therefore, expected. Then, the waveform missing phenomena happens inevitable. This phenomena happen not only by harmonic or subharmonic relation of Ft and Fs.
In the coherent sampling, there is the relationship described by EQU Ft/Fs=M/N
M and N are relatively prime integers, Ft is the input signal's repetition frequency, Fs is the sampling clock rate, N is the number of samples during a cycle of the input waveform to be measured, and M is the number of cycles of the input waveform to produce data of N different time points representing one cycle of the input waveform.
When the number of acquisition data N (=MFs/Ft) becomes smaller than the number of required interpolation data, which is Tw/Tres, the waveform missing phenomena are caused. Two measures for solving the phenomena are known as follows.
(1) In the first method of the random sampling, accumulated fluctuations (or random walks) of the input signal's repetition time period 1/Ft increase with the lapse of time by setting a long hold time duration Tmh, in which, for example, a time interpolator called the dual slope type is employed.
(2) In the second method of the random sampling, a sampling clock phase is randomly shifted every time after interpolation.
In the prior art 8, the second method is disclosed, in which the clock phase is randomly shifted by compulsive control.
So, the operation of data acquisition over the whole waveforms can be executed without being influenced with the fluctuations of the input signal's repetition frequency Ft. It requires several tens or hundreds clocks periods or more until the shifted clock phase converges and the clock cycle time becomes constant. Accordingly, the second method is better than that of the first from the viewpoint of the waveform missing phenomena. In the shortening the hold time duration Tmh, the improvement is not, however, enough for convenient use.
In the first method, the long hold time duration Tmh is originally set. It is not, therein, intended to obtain a short hold time duration.
In FIGS. 7, 8, 9 and 10, reproduced waveforms (replicas) are shown by simulation. The simulation is executed by the first method of the random sampling setting the short or long hold time duration Tmh. Each of the clock cycle times Tcs being set causes the constant holdoff system to trouble with the waveform missing phenomena. In those sampling operations, data acquisitions are executed 640 times, i.e., 10 times per a time resolution Tres=Tc/64, and each of signal waveforms has a mean cycle time of 10.
Each of those sampling is operated by the hold time duration Tmh, the clock cycle time Tc, the starndard deviation of the fluctuation FL, and the number M (=64) of data to be displayed per Tc.
In FIG. 7, Tmh=10.3(Tc), Tc=10)1+8/M), and FL=0.001
In FIG. 8, Tmh=1000.3(Tc), Tc=10(1+8/M, and FL=0.001
In FIG. 9, Tmh=1000.3(Tc), Tc=10(1+32/M), and FL=0.0003
In FIG. 10, Tmh=50000.3(Tc), Tc=10(1+32/M), and FL=0.00003
The waveform missing phenomena can be found in FIG. 7 of Tmh=10.3(Tc). In FIG. 8 employing longer Tmh=1000.3(Tc) than that of FIG. 7, no waveform missing phenomena can be found. By changing the fluctuation FL=0.001 of FIG. 8 to a smaller FL=0.0003 of FIG. 9, the waveform missing phenomena appears. By change Tmh=1000.3(Tc) of FIG. 9 to a longer Tmh=50000.3(Tc), the waveform missing phenomena can not be found again.
In FIGS. 7 to 10 showing random sampling simulations, when the starndard deviation of the input signal fluctuation FL becomes small, it becomes difficult sometimes to execute data acquisition over the whole waveform. And the waveform missing phenomena can be observed. Happenings of the phenomena are avoidable with a long hold time duration Tmh.
However, the measures using the long hold time duration are impractical. Because a long period of time is needed to acquire waveform data.
There are pretrigger and posttrigger acquisition processes in the random sampling. The pretrigger acquisition process is for acquiring signal data before a trigger input. The posttrigger acquisition process is for acquiring signal data after the trigger input.
In both processes, each of time bins corresponding to each of points on the signal waveform stores each of data of the signal waveform. It is the cause of waveform missing phenomena that some of time bins remain vacant.
FIG. 11 shows the operating of the pretrigger and posttrigger acquisition processes.
The signal 1 of FIG. 11 (a) is to be measured. The signal 1 is sampled by the sampling clock 3 at time points t109 to t114 to obtain data D1 to D6.
The synchronizing signal 52 of FIG. 11 (b) is selected for reference from triggers synchronizing with the input signal 1 of (a) to be measured.
The synchronizing signal 52 of (b) occurs at the signal 1 rising edge t100. The synchronizing signal 52 can be picked off from the signal 1. In the hold state HO of (d), no trigger is acceptable. The data D1 is obtained by sampling the signal 1 at the time point t109 of the hold state HO. The data D1 is stored into the acquisition memory 30 of (e) which is FIFO (first in first out memory).
The memory 30 of FIG. 11 consists of 3 data words for convenience' sake.
The data D2 sampled at the time point t110 is stored into the acquisition memory 30 of 3 words. The first and second words are respectively occupied with the data D2, D1. But, the third word is vacant at t110.
At the time appoint t102, the hold signal HO of FIG. 11 (d) turns from the hold state to inhibit triggering to the unhold state HOF to be triggerable.
Although it is in unhold state HOF, no trigger is applied. So, at the time point t112, the signal 1 of (a) is sampled by the clock 3 of (c) to obtain the data D4. The data D4 is stored into the acquisition memory 30 of (e). At the time point t112, the acquisition memory 30 has been already fully occupied.
The oldest data D1 is, therefore, abandoned and the data D4, D3 and D2 are stored into the memory 30.
At the time point t100, a synchronizing signal 52 of FIG. 11 (b) is applied and the unhold state HOF is turned to the hold state to inhibit triggering. The signal 52 synchronizes with the signal 1 of (a).
At the time points t113, the signal 1 of (a) is sampled by the clock 3 of (c) to obtain the data D5 stored into the acquisition memory 30 of (e). In the memory 30, the oldest data D2 is abandoned and the data D5, D4 and D3 are stored into the memory 30. In like manner, at the time point t114, the data D6, D5 and D4 are stored into the memory 30 of (e).
A time window of FIG. 11 (g) is preset before the measurement. The time window of (g) consists of two time periods for the pretrigger and posttrigger acquisition processes. In the pretrigger acquisition process, the data before the synchronized signal 52 at the time point t100 are acquired. In the posttrigger acquisition silicon process, the data after the synchronizing signal 52 at the time point t100 are acquired.
In FIG. 11, the acquisition memory of three words are employed as an example. One of the three words is assigned to the pretrigger acquisition process. The remaining two words are assigned to the posttrigger acquisition process. Then, the data D4 is stored into a time bin, which is one of five time bins assigned to the pretrigger acquisition process. Each of time bins corresponds to the time resolution Tc/5.
The data D5 and D6 are stored as the posttrigger acquisition. When two words of data D5 and D6 are stored into time bins of the waveform memory 50, the operation of the data acquisition ends.
In FIG. 11 (f), contents of the waveform memory 50 are shown. The memory 50 consists of 15(=3.times.5) time bins for 3Tc, in which the time resolution is Tc/5. There is a time difference Tsn between the time point t100 of the synchronizing signal 52 and the time point t113 of the sampling clock 3 just after t100. The data D6, D5 and D4 acquired in the memory 30 at the time point t114 are respectively stored into bins corresponding to the time difference Tsn in FIG. 11 (f). The data D4 is posted by the clock cycle time Tc before the data D5. The data D6 is posted by the clock cycle time Tc after the data D5.
Data of five time bins are obtained during the pretrigger acquisition processes. In the time window of FIG. 11 (g), only the data D4, which is acquired during t109 to t100 of one pretrigger acquisition process, is representatively shown. In like manner, data of ten time bins are obtained during the posttrigger acquisition processes. In the time window of FIG. 11 (g), only the data D5 and D6, which are acquired during t100 to t114 of one posttrigger acquisition process, are representatively shown.
In the following pretrigger and posttrigger acquisition processes after the next synchronizing signal 52 not shown, different points from points D1 to D6 of (a) will be sampled to obtain other data. Those data will be respectively stored into time bins of the waveform memory 50 of (f) corresponding to sampled points. The above-mentioned operations are repeated the predetermined times. Thus, the data of the fifteen time bins are displayed. In case that all of the fifteen time bins are occupied with data, the waveform missing phenomena can not be caused.
In FIG. 12, there are shown time charts of the equivalent time sampling for displaying are produced waveform (replica). In (a), the sampling clock 3 is shown, In (b), the sampling clock 3' following the clock 3 of (a) is shown. A signal 1 to be measured is shown in (c). The reference triggers RTs are generated at the rising edge of signal waveforms displayed with thick lines. The reference triggers RTs are used as the synchronizing signal 52 not shown.
The sample point SP1-1 on the signal 1 of (c) is sampled by the clock 3 of (a) at the clock timing c1 just after the reference trigger RT1 of the signal 1 of (c). The data of the sample point SP1-1 is acquired and displayed as one instantaneous value of the reproduced waveform (replica) of (d). In like manner, the sample point SP1-2 is sampled at c2, the sampled data is acquired and displayed as (d).
The clock timing c13 of (a) corresponds to c1' of (b). The data of the sample point SP2-1 is sampled by the clock 3 at the clock timing c13 (or c1') just after the reference trigger RT2. The data of the sample point SP2-1 is acquired and displayed as another instantaneous value of the reproduced waveform (replica) of (d). The data of SP2-2 is sampled at c14 (or c2') to be dis played as shown in (d). Four data of SP1-1, -2, SP2-1 and -2 are acquired from two signal waveforms of (c). The reproduced waveform (replica) of (d) is equivalent to data acquired from a single waveform.
The rising edge of the displayed waveform (replica) of (d) corresponds to the reference trigger RT1 or RT2. The time difference Ts1 is between the reference trigger RT1 and the clock timing c1 just after RT1, and Ts2 is between RT2 and c13 (or c1') just after RT2.
The reference trigger time period Trr1 between RT1 and RT2 is described by EQU Trr1=12Tc+Ts1-Ts2=12Tc-.delta. Ts1
in which EQU .delta. Ts1=Ts2-Ts1
In the same manner, the reference trigger time period Trr2 between RT2 and RT3 is described by EQU Trr2=12Tc+Ts2+Tc-Ts3=12Tc-.delta. Ts2+Tc
in which EQU .delta. Ts2=Ts3-Ts2
In the timings of FIG. 12, .delta. Ts1 and .delta. Ts2 are respectively given by EQU .delta. Ts1=Ts2-Ts1=-(1/5)Tc EQU .delta. Ts2=Ts3-Ts2=(4/5)Tc
wherein as if .delta. Ts1 and .delta. Ts2 are unequal. However, .delta. Ts2-Tc=-(1/5)Tc. If .delta. Ts2-Tc is anew replaced to .delta. Ts2, the new .delta. Ts2 is equal to .delta. Ts1. This is easily understandable from the fact of Trr1=Trr2 in FIG. 12.
In FIG. 13, there are shown relationships between the reproduced waveforms (replica) of FIG. 12 (d) and many time bins which store data of sample points SP referring the reference trigger RT. The signal 1 and the clock 3 are, therein, set in their cycle time so as to obtain the relation of EQU .delta. Ts=-(1/5)Tc
Upward arrows indicate clock timings cs of the clock 3 (or 3'). Horizontal arrows show time differences Tss. Eight time bins are provided for a clock cycle time Tc. Namely, the time resolution is Tc/8. When all of the eight time bins are occupied with waveform data, no waveform missing phenomena occurs.
With the lapse of time, seven reference triggers RT1 to 5, RT1' and 2' are continuously generated. Reference triggers RT4, 5, 1' and 2' are not shown in FIG. 12. They are continuously generated after the RT3.
The black dots on the signal 1 are sampled points representing instantaneous amplitudes of the signal 1. The sampling are executed by the clock 3 generated during a period of about 1.5Tc after each of the seven reference triggers RTs. The seven black dots obtained by the equivalent time sampling reproduce a waveform as the input signal 1.
In the example of FIG. 13, the clocks 3 of the reference triggers RT1 to 5 sample different time points on the signal 1. But the clock 3 of the reference trigger RT1' samples at relatively the same time point as that of RT1, and the clock 3 of RT2' samples at relatively the same time point as that of RT2. The same time points as the previous series of samples on the signal 1 are repeatedly sampled. In spite of the continuation of the sampling, it is difficult to obtain more sample points than five.
In FIG. 13, the time resolution for acquiring sampled data is Tc/8. There are employed eight time bins corresponding the clock 3 of the reference trigger RT1. The clock cycle time Tc is resolved by eight time bins. Five time bins with circles are occupied with data sampled by the clocks 3 of RT1 to RT2'. The other three time bins are vacant, namely, the waveform missing phenomena can be observed.